Tests for a given linear structure of the mean direction of the langevin distribution

1992 ◽  
Vol 44 (1) ◽  
pp. 147-156 ◽  
Author(s):  
Yoko Watamori
Keyword(s):  
Linear Structure ◽  
The Mean ◽  
Langevin Distribution

scholarly journals Bakerian Lecture: Stereochemical types and valency groups

1940 ◽  
Vol 176 (965) ◽  
pp. 153-180 ◽  
Keyword(s):  
Linear Structure ◽  
Trigonal Prism ◽  
Experimental Results ◽  
The Other ◽  
Trigonal Bipyramid ◽  
Distorted Tetrahedron ◽  
Electron Pairs ◽  
Regular Octahedron ◽  
The Mean ◽  
Tellurium Tetrachloride

The arrangements in space of the covalencies of a polyvalent atom , while they are subject to small variations seldom exceeding 5 or 10°, tend to conform to quite a limited number of types. It is desirable to relate this grouping to some familiar property of theatom . The property here used is the size (in G. N. Lewis’s sense) of the valency group of the centralatom , and the number of shared electrons which it contains, together with that of the preceding (unshared) electronic group in the atom . The experimental results show the following relations. I. When the valency group is less than 8 we have with a covalency of 2 a linear structure (as in Cl-Hg-Cl), and with one of 3 a plane with equal angles of 120° (as in BF 3 ). II. With a complete octet the arrangement can be either tetrahedral or planar. When the covalency is less than 4 it is always derived from the tetrahedron, as in the triangular OH 2 and th e pyramidal NH 3 . The fully shared octet is always tetrahedral when the preceding group ( n in the grouping ( n ) 8) is 2, 8, or 18. In the transitional elements where 8 < n < 18, it is tetrahedral if n is not much more than 8, and planar if it is not much less than 18; but the two series overlap. III. When there are 10 valency electrons, at least 2 of them (the “inert pair”) unshared, the structure with a dicovalent atom (as in M[I 3 ]) is linear: that of a 4-eovalent atom is found in the thallous and plumbous salts to be planar, but in tellurium tetrachloride it may be a distorted tetrahedron. IV . The peculiar 4-covalent duodecet in M[ICl 4 ] is planar. V. A covalency of 5 is always found to give a trigonal bipyramid. VI. Covalency 6. Three structures are theoretically possible, a trigonal prism, a trigonal antiprism , and a regular octahedron. Experimentally the octahedron is always found, except in a few giant molecules such as those with a nickel-arsenide lattice. The regular octahedron has been found with practically every possible size of the preceding group, as well as with the “inert pair” of electrons. VII. A covalency of 7 can have two different structures, one derived from the octahedron and the other from the trigonal prism. VIII. Covalency 8. The only com pound exam ined, K 4 [Mo(CN) 8 ], has a dodecahedral arrangement of the 8 CN groups. Nearly (but not quite) all the structures can be even more simply related to the size of the valency group by assuming that the mean positions of the electron pairs in this group are the same whether they are shared or not, the structure being linear for 4 electrons, plane symmetrical for 6, either tetra­hedral or plane for 8, a trigonal bipyramid for 10, and an octahedron for 12.

Photo-electric Hβ and uvby photometry

1966 ◽  
Vol 24 ◽  
pp. 170-180
Author(s):  
D. L. Crawford
Keyword(s):  
Narrow Band ◽  
Line Strength ◽  
Interference Filter ◽  
B Stars ◽  
Relative Line ◽  
The Mean ◽  
F Stars ◽  
Two Parameters ◽  
Filter Photometry ◽  
The Continuum

Early in the 1950's Strömgren (1, 2, 3, 4, 5) introduced medium to narrow-band interference filter photometry at the McDonald Observatory. He used six interference filters to obtain two parameters of astrophysical interest. These parameters he calledlandc, for line and continuum hydrogen absorption. The first measured empirically the absorption line strength of Hβby means of a filter of half width 35Å centered on Hβand compared to the mean of two filters situated in the continuum near Hβ. The second index measured empirically the Balmer discontinuity by means of a filter situated below the Balmer discontinuity and two above it. He showed that these two indices could accurately predict the spectral type and luminosity of both B stars and A and F stars. He later derived (6) an indexmfrom the same filters. This index was a measure of the relative line blanketing near 4100Å compared to two filters above 4500Å. These three indices confirmed earlier work by many people, including Lindblad and Becker. References to this earlier work and to the systems discussed today can be found in Strömgren's article inBasic Astronomical Data(7).

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

the effects of resonance with Jupiter in the system of short-period comets

1974 ◽  
Vol 22 ◽  
pp. 193-203
Author(s):  
L̆ubor Kresák
Keyword(s):  
Structural Effects ◽  
Order Resonance ◽  
Mean Motion ◽  
Short Period ◽  
The Mean ◽  
Compound Effect ◽  
Low Order

AbstractStructural effects of the resonance with the mean motion of Jupiter on the system of short-period comets are discussed. The distribution of mean motions, determined from sets of consecutive perihelion passages of all known periodic comets, reveals a number of gaps associated with low-order resonance; most pronounced are those corresponding to the simplest commensurabilities of 5/2, 2/1, 5/3, 3/2, 1/1 and 1/2. The formation of the gaps is explained by a compound effect of five possible types of behaviour of the comets set into an approximate resonance, ranging from quick passages through the gap to temporary librations avoiding closer approaches to Jupiter. In addition to the comets of almost asteroidal appearance, librating with small amplitudes around the lower resonance ratios (Marsden, 1970b), there is an interesting group of faint diffuse comets librating in characteristic periods of about 200 years, with large amplitudes of about±8% in μ and almost±180° in σ, around the 2/1 resonance gap. This transient type of motion appears to be nearly as frequent as a circulating motion with period of revolution of less than one half that of Jupiter. The temporary members of this group are characteristic not only by their appearance but also by rather peculiar discovery conditions.

Contrast of Filament Bright Rims

1994 ◽  
Vol 144 ◽  
pp. 365-367
Author(s):  
E. V. Kononovich ◽  
O. B. Smirnova ◽  
P. Heinzel ◽  
P. Kotrč
Keyword(s):  
High Altitude ◽  
Line Profile ◽  
Line Center ◽  
Magnetic Structures ◽  
The Mean

AbstractThe Hα filtergrams obtained at Tjan-Shan High Altitude Observatory near Alma-Ata (Moscow University Station) were measured in order to specify the bright rims contrast at different points along the line profile (0.0; ± 0.25; ± 0.5; ± 0.75 and ± 1.0 Å). The mean contrast value in the line center is about 25 percent. The bright rims interpretation as the bases of magnetic structures supporting the filaments is suggested.

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